The principal of Higginson-Lewis K-8 School and one of her first-grade teachers stood amid a swirl of school-shopping families at the Showcase of Schools, waiting to deliver their sales pitch.

Moms and dads barely glanced at the Roxbury school’s display. Only one man made a beeline for the table - and he works at the school. Families that did congregate nearby were just spillover from the crowd waiting to speak with representatives of the “premier schools,’’ as one father put it.

“It’s like being a Hilton Hotel in between two Ritzes,’’ Simmons, the first-grade teacher, said of the schools to her right and left, Hernandez K-8 and Kilmer K-8, both with more applicants than prekindergarten seats. The inverse is true at Higginson-Lewis, making it one of the least sought-after schools in Boston - at least according to a school district tally akin to a judge’s score sheet.

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The city uses a lottery system that was intended to give all students access to high-achieving classrooms, regardless of neighborhood or life circumstance. But families fixate on a collection of well-known, fiercely sought-after schools, largely ignoring those with lesser reputations. And over the past two decades, popularity has often become a proxy for quality, making it even harder for schools to get off that second rung.

Popularity is driven by parents with time, inclination, and sometimes the means to enter the school lottery early, armed with information and expectations. Their preferences create a system of prized schools, and those in low demand - schools whose reputations have suffered because they are in higher-crime neighborhoods, serve predominantly poor students, and have, in some cases, test scores lower than average.

This stubborn asymmetry in demand makes it that much more difficult for Boston to achieve its goal of school equity and avoiding racial segregation.

The knowing parents who largely drive the process also tend to be those able to afford options outside the system if their child is shut out of the school of their choice. For them, it tends to be top choice, or nothing - and their exit from the system deprives the district of their engagement, fund-raising prowess, and the boost in profile that tends to come from having middle-class parents in a school.

At the popular Hurley K-8 in the South End, parents raised hundreds of thousands of dollars to subsidize arts and science education. The parents have a website that not only includes a wish list but also addresses this question: “I’m still considering private schools or moving to the suburbs for the public schools. Why should I consider the Hurley?’’

Principals at some of the city’s least-selected schools understand why district administrators want them to court middle-class families. But they worry that doing so may jeopardize their standing as advocates for the children who need them most, many of whom are black and Latino, from low-income families, and, in many cases, still learning English.

Starting in June, the Globe visited three of the district’s least-selected schools, watching eighth-graders graduate and transition to high school, speaking with kindergartners on the first day of school, listening to lessons, and witnessing the occasional meltdown. These are neither blue-ribbon schools nor those deemed to be in a state of crisis and thus receiving additional resources and intervention.

These schools are overlooked time and time again even though some have amenities that ought to be a draw - including pools and performance studios, which are sometimes lacking in more popular schools.

To walk the halls in overlooked schools is to pass classes with motivated teachers and focused students. Prekindergartners learn measurements and scientific theories by making pizza. They sort brightly colored blocks by shape - hexagon, octagon, diamond, and triangle.

“None of that,’’ Oliver said, “is going to trump the fact of where we are or who we are.’’

Each year, the district creates a “demand report’’ to help inform parents’ decisions. It shows how many parents listed a school among their top three picks. Parents look at the list and seize on schools they like, but also immediately see the schools they want to avoid, schools they often know little about.

Miranda Webster, a 32-year-old event planner at Boston University, referred to the report while researching school options for her daughter during the November Showcase of Schools, an event at which principals, teachers, and parents promote their schools. The Jamaica Plain mother stopped to take in the tableau of options before her.

“It’s so sad to see all the schools that nobody is interested in are in Roxbury and Dorchester,’’ Webster said, looking at the spreadsheet. “They are just being left behind.’’

Webster wasn’t considering these schools for her soon-to-be 4-year-old, but she wouldn’t be opposed to visiting a few, “just because I’m curious.’’ And she did: Later, she went to the Mendell Elementary on the edge of Roxbury.

The more she researches, the more her definition of quality changes. A gut feeling now trumps proximity to home. And an up-and-coming parent group beats out an established organization that might be hard to break into.

“It’s such a crazy system. You have to be open to other options,’’ she said, before giving the Higginson-Lewis table a once over and moving on.

Who chooses what

On paper, these are the schools nobody wants to attend.

In the demand report, each has fewer than one applicant per seat. One of them, Marshall Elementary, has a 0.30-applicant-per-seat ratio for prekindergarten, the lowest in the city. But none of the Dorchester school’s classrooms are empty. In fact, there’s a waiting list for its 13 prekindergarten and kindergarten classes.

So, if all of the seats are full and there is a list of students waiting, the question becomes: How is it that these are under-chosen schools?

The answer lies in who is, and who is not, choosing a school and when they choose. Popular schools have become synonymous with the choices of white middle-class families, principals and families say. And the demand report reflects the choices of families who choose early.

Oliver said parents of color and those in low-income communities “don’t always go in to make choices when the lottery starts. We have a lot of people who can’t make a commitment until June or even Labor Day.’’

At Edison K-8, a Brighton school with a transient population of students from 65 countries, there is still a list of students waiting to fill prekindergarten seats. “Someone leaves and someone comes in,’’ said Mary Driscoll, the school principal. Her school, on the city’s western edge, is one of the least-selected in the north assignment zone. Students enroll from East Boston and the South End. Some are in Boston temporarily for medical treatment, or their parents are international graduate students, or they are homeless.

Even though there is a waiting list, the school only has a 0.77- applicant-per-seat ratio for pre-K on the demand report. “We might not have a lot of first choices but a lot of second, third, or fourth choices,’’ Driscoll said.

The lottery system traces its roots to a federal court decision nearly four decades ago. In 1974, federal Judge W. Arthur Garrity ordered students bused from one neighborhood to another in the name of desegregation and in response to the city’s absolute failure to provide quality education for all, particularly for minority pupils. In the mid-1980s, the federal courts returned control of the schools to the city with one important stipulation: no return to neighborhood schools and the inequality they spurred.

The lottery system was created in the name of giving parents more choice. Still, Boston’s dreams of equal access to quality remain deferred, with many of the least-selected schools lacking racial and economic diversity. The Higginson-Lewis has only 10 white students in a school of about 425, and Marshall has just eight white students in a school of 713.

“People will come to visit and they will say: ‘How many white students are in the class? I don’t want my child to be the only one,’ ’’ said Oliver, the Higginson principal.

Few children travel from West Roxbury, Roslindale, or Hyde Park - all within the school’s attendance zone. Instead, most - about 65 percent - live within a 1 1/2-mile radius.

As if on cue, Oliver pulls open the door to a classroom holding a citywide program for autistic children. Inside are 10 who struggle with social interaction and communication and are learning to conduct a proper introduction. The teacher sees Oliver’s visit as the perfect time to practice.

One by one, the students approach. Some shake hands. Others offer only their names. Some do both. One student seizes the opportunity to explain what he has just learned about animals in sub-Saharan Africa.

The school educates about 40 children in five classes for autistic children. Only four are white.

“You kind of have to wonder, why is that? It’s a citywide program,’’ Oliver said back in the hallway. “You would think . . . there would be a mix [of students] so, I have to ask: Are there only black and brown children with autism?’’

The answer, of course, is no. White children are actually more likely to be diagnosed with autism.

“This could be the castle on the hill, but some people will never pick this place,’’ she said.

Middle-class parents often aren’t willing to send their children to a school next to Malcolm X Park in Roxbury or on a street sandwiched between Geneva Avenue and Bowdoin Street in Dorchester, where neighborhood violence has, at times, landed on the school’s doorstep.

Manny DaSilveira’s family lived within walking distance of Marshall Elementary until November, when they moved, and his sons, in kindergarten and first grade, enrolled in Winthrop Elementary in Dorchester, another under-selected school.

“The experience over there at the Marshall was actually really good,’’ DaSilveira said, adding that they are happy with Winthrop thus far. “I went into the class and stayed often. The teachers were excellent. Unfortunately, we had to move.’’

When looking to enroll DaSilveira’s son Joseph in kindergarten, Joseph’s mother researched schools with better reputations but decided the relationship they had with Marshall and its proximity to home made it the right fit. Feedback from other parents is important, but, DaSilveira said, witnessing a teacher manage a classroom and the principal manage the school is critical too, which is exactly what administrators encourage. “The best way to find out is to go out and speak with teachers and talk to the people in charge and also be in the classrooms.’’

Marshall’s principal, Teresa Harvey-Jackson, knows about the intersection of perception and reality. She has walked the halls of Marshall for 18 years as principal and seen the neighborhood change, watching violence ebb and flow. Every day, when students enter the school, a backpack full of urban ills comes with them.

She knows each child’s story - the girl who begs her to call home and urge mommy to return daddy’s keys so he won’t beat mommy again, the homeless boy acting out because he hasn’t seen his mother in a week, first-grade twins who missed 90 days of classes. And though Harvey-Jackson empathizes with the heavy burdens carried on such slight shoulders, it’s her job to educate, not pity students.

“If they’re ever going to have a real future, it’s through education,’’ she said. “If we cry over these issues, we can’t help them.’’

Moving at a quick clip through the halls of the massive school, Jackson walks room to room, monitoring behavior, observing teachers, engaging students.

“Are you working today, and can you get the pencil so we’ll have something to work with?’’ she asks a pouting fourth-grader.

No response. She persists: “Are you reading? Yes or no.’’

A shoulder shrug.

“That’s not an answer.’’

His teacher chimes in: “He’s hungry.’’

The boy missed breakfast at home and at school.

Down the hall, his schoolmates are plugging along with math, working on data sets and graphs. “We did a lot last week,’’ offered second-grader Josiah Fuentes. “That’s why we know a lot.’’

As she makes her rounds, listening to lessons - properties of homemade Play-Doh, bug classifications, and women in space - Harvey-Jackson passes the school’s peace garden. It’s a reminder that everyone matters, and of what lies between the Marshall and the reputation it deserves. Surrounding the base of a tree are bricks painted with the names of former students, all felled by violence.

Winners and losers

School choice is “pretty complicated stuff, and people are always eager to come up with pretty simple solutions,’’ said Curt Dudley-Marling, a Boston College professor who studies patterns of school failure and success. “It always seems to me that it’s rigged for parents who have the most resources.’’

Not all families have the benefit of active parent groups that organize school tours to help families vet their options, which in Boston could mean as many as 20 public school options, not including charters. Single parents, families new to the country, parents of disabled children, or families struggling with the demands of life often are unable to investigate every option.

“I can’t imagine they have time, much less the resources, to go to fairs and all these things,’’ Dudley-Marling said. Instead, they, like most people, default to what they have heard within their circle of influence.

Chatting as they wait for their children is a daily ritual for cousins Vimbai Bush and Elizabeth Tyler. It is one that almost didn’t come to pass at all.

Bush toyed with enrolling her son Zafir in a different school last year when he started prekindergarten. But she didn’t get very far before deciding against it. With two older children having already graduated from Higginson-Lewis, Bush had a relationship with the school.

When the mother of five was evicted from her home, she knew her son would stay on the “straight and narrow’’ academically, because of support from the school.

People in the neighborhood take care of one another, Bush said, before fussing at her cousin’s 9-year-old daughter for interrupting.

Tyler said Higginson-Lewis was the only option for her daughter. It is an integral part of their community. The cousins threaten to tell parents and the principal when they catch middle-schoolers skipping class at McDonald’s. “We know them,’’ she said, “and they listen.’’

Prospects for change

Boston is seeking to overhaul its school assignment process, but city officials are adamant that parents will always have some degree of choice. “The question,’’ said Superintendent Carol R. Johnson, “is how much choice is enough and what are the . . . choices parents have?’’

The district received a $240,000 federal grant two years ago to find answers to that question after it tried unsuccessfully to change the process. Parents, educators, and civil rights activists argued that the plan under discussion, to move from three to five assignment zones, left too many low-income students with fewer opportunities to attend high-quality schools.

No new plan has emerged, and the grant was extended.

Johnson said before school assignment changes can occur, the district must address quality and the perception of it.

“Parents listen to other parents,’’ she said. “On the other hand, parents rely on quantitative data.’’

Improvements, Johnson said, will drive parent demand, offering Orchard Gardens K-8 in Roxbury as an example. The school, plagued with chronically low standardized test scores, was overhauled a year and a half ago, getting new administrators, teachers, and programs. And while test scores are still low, they are much higher than a year ago.

And word of the school’s progress is spreading. A group of South End parents - traditionally outside of the school’s demographic - visited during a recent open house. And while those families haven’t chosen Orchard Gardens yet, that they showed up is heartening, Johnson said.

“We are simultaneously trying to improve the quality of schools in our district and make every family feel like they don’t have one choice but they have multiple choices,’’ the superintendent said.

Stephanie Shapiro Berkson, a South End mom with unabashed faith in advice given from the online parenting group Garden Moms, knows she has options. But three schools keep popping to the top of her list. This is her second time going through the lottery; her 5-year-old son remains on waiting lists at Hurley, Eliot in the North End, and Quincy in Chinatown - all extremely popular, high-achieving schools. But another school, Mission Hill K-8, is nudging onto her list.

She and her husband, an orthopedic surgeon, recently attended the North Zone Showcase of Schools looking for what she called “hidden gems’’ to round out her list. She flirted with several under-chosen schools, liking the Edison’s diversity, new outdoor classrooms, and state-of-the-art playground but hating its location, size, and what she considered a lackluster parent group. She was unimpressed when she heard that only about 20 moms and dads came to one of the parent organization’s recent meetings.

“And it’s somewhere in Brighton . . . it had nothing to do with our neighborhood,’’ said Shapiro Berkson, a public health doctoral student.

The mother of two is committed to public schools and the idea that they foster social change. It is mothers like her, who know how to work the system, who are needed to turn schools around, she acknowledged. So she also looked at Blackstone Elementary, in her neighborhood. “I was on a mission to maybe put Blackstone on my list, but six people from everywhere told me not to.’’

So she is conflicted, torn between what she believes in, and what she won’t do - gamble with her child’s future.

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